Alright, let's break down the solution step-by-step to determine the probability that both events A and B will occur when two six-sided dice are tossed.
### Step 1: Determine the Probability of Event A
Event A states that the first die lands on a 1 or a 2. Since a six-sided die has six faces, the total number of possible outcomes for a single toss is 6.
- The favorable outcomes for Event A are the first die showing either a 1 or a 2.
- Therefore, there are 2 favorable outcomes for Event A.
The probability of Event A, [tex]\( P(A) \)[/tex], is calculated as:
[tex]\[ P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{2}{6} = \frac{1}{3} \approx 0.333 \][/tex]
### Step 2: Determine the Probability of Event B
Event B states that the second die lands on a 5. Similar to the first die, the second die also has 6 faces.
- The favorable outcome for Event B is the second die showing a 5.
- Therefore, there is 1 favorable outcome for Event B.
The probability of Event B, [tex]\( P(B) \)[/tex], is calculated as:
[tex]\[ P(B) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{1}{6} \approx 0.167 \][/tex]
### Step 3: Calculate the Probability of Both Events Occurring
Since events A and B are independent (the outcome of one die does not affect the outcome of the other), the probability of both events occurring simultaneously, denoted as [tex]\( P(A \text{ and } B) \)[/tex], is the product of the individual probabilities.
[tex]\[ P(A \text{ and } B) = P(A) \times P(B) \][/tex]
[tex]\[ P(A \text{ and } B) = \left(\frac{1}{3}\right) \times \left(\frac{1}{6}\right) = \frac{1}{18} \approx 0.056 \][/tex]
So, the probability that both events A and B will occur when two six-sided dice are tossed is approximately [tex]\( 0.056 \)[/tex], or to be precise [tex]\( \frac{1}{18} \)[/tex].
